SCALABLE FRAMES GITTA KUTYNIOK, KASSO A. OKOUDJOU, FRIEDRICH PHILIPP, AND ELIZABETH K. TULEY Abstract. Tight frames can be characterized as those frames which possess optimal numerical stability properties. In this paper, we consider the question of modifying a general frame to generate a tight frame by rescaling its frame vectors; a process which can also be regarded as perfect preconditioning of a frame by a diagonal operator. A frame is called scalable, if such a diagonal operator exists. We derive various characterizations of scalable frames, thereby including the infinite-dimensional situation. Finally, we provide a geometric interpretation of scalability in terms of conical surfaces. 1. Introduction Frames have established themselves by now as a standard notion in applied mathematics, computer science, and engineering, see [9, 7]. In contrast to orthonormal bases, typically frames form redundant systems, thereby allowing non-unique, but stable decompositions and expansions. The wide range of applications of frames can be divided into two categories. One type of applications utilize frames for decomposing data. Here typical goals are erasure-resilient transmission, data analysis or processing, and compression – the advantage of frames being their robustness as well as their flexibility in design. A second type of applications requires frames for expanding data. This approach is extensively used in sparsity methodologies such as Compressed Sensing (see [11]), but also, for instance, as systems generating trial spaces for PDE solvers. Again, it relies on non-uniqueness of the expansion which promotes sparse expansions and on the flexibility in design. All such applications require the associated algorithms to be numerically stable, which the subclass of tight frames satisfies optimally. Thus, a prominent question raised in several publications so far is the following: When can a given frame be modified to become a tight frame? The simplest operation to imagine is to rescale each frame Date: April 11, 2012. 2000 Mathematics Subject Classification. 12D10, 14P05, 15A03, 15A12, 42C15, 65F08. 1 2 G. KUTYNIOK, K. A. OKOUDJOU, F. PHILIPP, AND E. K. TULEY vector. Therefore this question is typically phrased in the following more precise form: When can the vectors of a given frame be rescaled to obtain a tight frame? This is the problem we shall address in this paper. 1.1. Tight Frames. Let us first state the precise definition of a frame and, in particular, of tight and Parseval frames to stand on solid ground for the subsequent discussion. Letting H be a real or complex separable Hilbert space and letting J be a subset of N, a set of vectors Φ = {ϕj }j∈J ⊂ H is called a frame for H, if there exist positive constants A, B > 0 (the lower and upper frame bound) such that X (1) Akxk2 ≤ |hx, ϕj i|2 ≤ Bkxk2 for all x ∈ H. j∈J A frame Φ is called A-tight or just tight, if A = B is possible in (1), and Parseval, if A = B = 1 is possible. Moreover, if |J| < ∞ (which implies that H = KN with K = R or K = C), the frame Φ is called finite. To justify the claim of numerical superiority of tight frames, let Φ = {ϕj }j∈J
⊂ H be a frame for H and let TΦ : H → ℓ2 (J) with TΦ x := hx, ϕj i j∈J denote the associated analysis operator. Its adjoint TΦ∗ , the synthesis operator of Φ, maps ℓ2 (J) surjectively onto H. From the properties of TΦ , it follows that the frame operator SΦ := TΦ∗ TΦ of Φ, given by X SΦ x = hx, ϕj iϕj , x ∈ H, j∈J is a bounded and strictly positive selfadjoint operator in H. These properties imply that Φ admits the reconstruction formula X x= hx, ϕj iSΦ−1 ϕj for all x ∈ H. j∈J Inversion requires particular numerical attention, which implies that SΦ = const·IH is desirable (IH denoting the identity on H, for H = KN we will use IN ). And in fact, tight frames can be characterized as precisely those frames satisfying this condition. Thus an A-tight frame admits the numerically optimally stable reconstruction given by X x = A−1 · hx, ϕj iϕj for all x ∈ H. j∈J SCALABLE FRAMES 3 1.2. Generating Parseval Frames. This observation raises the question on how to carefully modify a given frame – which might be suitable for a particular application – in order to generate a tight frame. It is immediate that this question is equivalent to generating a Parseval frame provided we allow multiplication of each frame vector by the same value. Thus typically one seeks to generate Parseval frames. −1/2 A very common approach is to apply SΦ to each frame vector of a frame Φ, which can be easily shown to yield a Parseval frame. This approach is though of more theoretical interest due to the repetition of the problem to invert the frame operator. Hence, this construction is often not reasonable in practice. The simplest imaginable variation of a frame is just scaling its frame vectors. We thus coin a frame scalable, if such a scaling leads to a Parseval frame. It should also be pointed out that the scaling of frames is related to the notion of signed frames, weighted frames as well as controlled frames (see, e.g., [15, 1, 16]). It is evident that not every frame is scalable. For example, a basis in R2 which is not an orthogonal basis is not scalable, since a frame with two elements in R2 is a Parseval frame if and only if it is an orthonormal basis. As a first classification, the finite-dimensional version of Proposition 2.4 shows that a frame Φ in KN with analysis operator TΦ (the rows of which are the frame vectors) is scalable if and only if there exists a diagonal matrix D such that DTΦ is isometric. Since the condition number of such a matrix equals one, the scaling question is a particular instance of the problem of preconditioning of matrices. 1.3. An Excursion to Numerical Linear Algebra. In the numerical linear algebra community, the problem of preconditioning is wellknown and extensively studied, see, e.g., [8, 13]. The problem to design preconditioners involving scaling appears in various forms in the numerical linear algebra literature. The common approach to this problem is to minimize the condition number of the matrix multiplied by a preconditioning matrix – in our case of DTΦ , where D runs through the set of diagonal matrices. As shown for instance in [4], this minimization problem can be reformulated as a convex problem. However, as also mentioned in [4], algorithms solving this convex problem perform slowly, and, even worse, there exist situations in which the infimum is not attained. As additional references, we wish to mention [6, 2, 8, 14, 18] for preconditioning by multiplying diagonal matrices from the left and/or the right, [19, 10, 12] for block diagonal scaling and [17, 5, 20] for scaling in order to obtain equal-norm rows or columns. 4 G. KUTYNIOK, K. A. OKOUDJOU, F. PHILIPP, AND E. K. TULEY 1.4. Our Contribution. Our contribution to the scaling problem of frames is three-fold. First, these are the leadoff results on this problem. Second, with Theorem 2.7 we provide various characterizations of (strict) scalability of a frame for a general separable Hilbert space. In this respect, a particular interesting characterization derived in Theorem 2.7 states that a frame Φ in a Hilbert space H is strictly scalable if and only if there exists a frame Ψ in a presumably different Hilbert space K such that the coupling of the frame vectors of Φ and Ψ in H ⊕ K constitutes an orthogonal basis. And, third, Theorems 3.2 and 3.6 provide a geometric characterization of scalability of finite frames. More precisely, we prove that a finite frame in RN is not scalable if and only if all its frame vectors are contained in certain cones. 1.5. Outline. This paper is organized as follows. In Section 2 we focus on the situation of general separable Hilbert spaces. We first analyze when a scaling preserves the frame property (Subsection 2.1), followed by a general equivalent condition in terms of diagonal operators (Subsection 2.2). Subsection 2.3 is devoted to the main characterization of strict scalability of frames. In Section 3 we then restrict to the situation of finite frames. First, in Subsection 3.1, we derive a yet different characterization tailored specifically to the finite-dimensional case. Finally, this result is shown to give rise to a geometric interpretation of scalable frames in terms of quadrics (Subsection 3.2). 2. Strict Scalability of General Frames In this section, we derive our first main theorem which provides a characterization of (strictly) scalable frames. We wish to mention that this result does not only hold for finite frames, but in the general separable Hilbert space setting. 2.1. Scalability and Frame Properties. We start by making the notion of scalability mathematically precise. We further introduce the notions of positive and strict scalability. Positive scalability ensures that no frame vectors are suppressed by the preconditioning. The same is true for strict scalability, which in addition prevents numerical instabilities caused by arbitrarily small entries in the matrix representation of the diagonal operator serving as preconditioner. Definition 2.1. A frame Φ = {ϕj }j∈J for H is called scalable if there exist scalars cj ≥ 0, j ∈ J, such that {cj ϕj }j∈J is a Parseval frame. If, in addition, cj > 0 for all j ∈ J, then Φ is called positively scalable. If there exists δ > 0, such that cj ≥ δ for all j ∈ J, then Φ is called strictly scalable. SCALABLE FRAMES 5 Clearly, positive and strict scalability coincide for finite frames, and each scaling {cj ϕj }j∈J of a finite frame {ϕj }j∈J with positive scalars cj is again a frame. In the infinite-dimensional situation this might not be the case. However, if there exist K1 , K2 > 0 such that K1 ≤ cj ≤ K2 holds for all j ∈ J, then also {cj ϕj }j∈J is a frame, see [1, Lemma 4.3]. A characterization of when a scaling preserves the frame property can be found in Proposition 2.2 below. This requires particular attention to the diagonal operator Dc in ℓ2 (J) corresponding to a sequence c = (cj )j∈J ⊂ K, which is defined by
Dc (vj )j∈J := cj vj j∈J , (vj )j∈J ∈ dom Dc , where  dom Dc := (vj )j∈J ∈ ℓ2 (J) : (cj vj )j∈J ∈ ℓ2 (J) . It is well-known that Dc is a (possibly unbounded) selfadjoint operator in ℓ2 (J) if and only if cj ∈ R for all j ∈ J. If even cj ≥ 0 (cj > 0, cj ≥ δ > 0) for each j ∈ J, then the selfadjoint operator Dc is nonnegative (positive, strictly positive, respectively). Before we present the announced characterization, we require some notation. As usual, we denote the domain, the kernel and the range of a linear operator T by dom T , ker T and ran T , respectively. Also, a closed linear operator T between two Hilbert spaces H and K will be called ICR (or an ICR-operator), if it is injective and has a closed range, i.e., if there exists δ > 0 such that kT xk ≥ δkxk for all x ∈ dom T . We mention that the analysis operator of a frame is always an ICRoperator. The following result now provides a characterization of when a scaling preserves the frame property. Proposition 2.2. Let Φ = {ϕj }j∈J be a frame for H with analysis operator TΦ and let c = (cj )j∈J be a sequence of non-negative scalars. Then the following conditions are equivalent. (i) The scaled sequence of vectors Ψ := {cj ϕj }j∈J is a frame for H. (ii) We have ran TΦ ⊂ dom Dc and Dc | ran TΦ is ICR. Moreover, in this case, the frame operator of the frame Ψ is given by SΨ = (Dc TΦ )∗ (Dc TΦ ) = TΦ∗ Dc Dc TΦ , where TΦ∗ Dc denotes the closure of the operator TΦ∗ Dc . Proof. (i)⇒(ii). Assume that Ψ is a frame and denote its analysis operator by TΨ . Then, for x ∈ H, the j-th component of TΨ x is given by (TΨ x)j = hx, cj ϕj i = cj hx, ϕj i = (Dc TΦ x)j . 6 G. KUTYNIOK, K. A. OKOUDJOU, F. PHILIPP, AND E. K. TULEY Hence, TΨ = Dc TΦ . As dom TΨ = H, this implies ran TΦ ⊂ dom Dc . Since Φ is a frame, ran TΦ is a closed subspace. And since Ψ is a frame, there exist A′ , B ′ > 0 such that A′ kxk2 ≤ kDc TΦ xk22 ≤ B ′ kxk2 for all x ∈ H. In particular, for v = TΦ x ∈ ran TΦ we have kDc vk22 = kDc TΦ xk22 ≥ A′ kxk2 ≥ A′ kTΦ k−2 kvk22 , which shows that Dc | ran TΦ is an ICR-operator. (ii)⇒(i). Conversely, assume that ran TΦ ⊂ dom Dc and that Dc | ran TΦ is ICR. By the closed graph theorem and ran TΦ ⊂ dom Dc , the operator Dc | ran TΦ is bounded, which implies the existence of A′ , B ′ > 0 such that A′ kvk22 ≤ kDc vk22 ≤ B ′ kvk22 holds for all v ∈ ran TΦ . Setting v = TΦ x and noting that TΦ is bounded and ICR, we obtain constants A′′ , B ′′ > 0 such that A′′ kxk2 ≤ kDc TΦ xk22 ≤ B ′′ kxk2 holds for all x ∈ H. Consequently, Ψ is a frame. It remains to prove the moreover-part, i.e., that (Dc TΦ )∗ = TΦ∗ Dc . Since Dc TΦ is bounded, so is its adjoint (Dc TΦ )∗ . In addition, it is easy to see that TΦ∗ Dc v = (Dc TΦ )∗ v holds for all v in the dense subspace dom Dc . Hence, TΦ∗ Dc is bounded and densely defined. Its bounded closure thus coincides with (Dc TΦ )∗ .  It is evident that the operator Dc in Proposition 2.2 is in general unbounded. The following corollary provides a condition on the frame Φ which leads to necessarily bounded diagonal operators Dc in Proposition 2.2. We remark that lim inf j∈J shall be interpreted as lim inf j∈J, j→∞ , which is a proper definition, since J ⊂ N was assumed. As it is custom, we set lim inf j∈J to ∞ if J is finite. Corollary 2.3. Let Φ, Ψ and c be as in Proposition 2.2 and assume that lim inf j∈J kϕj k > 0. Then Ψ is a frame if and only if Dc is bounded and Dc | ran TΦ is ICR. In this case, we have SΨ = (Dc TΦ )∗ (Dc TΦ ) = TΦ∗ Dc2 TΦ . Proof. If Dc has the above-mentioned properties, then Ψ is a frame by Proposition 2.2. If Ψ is a frame, then there exists B > 0 such that for each x ∈ H we have X c2j |hx, ϕj i|2 ≤ Bkxk2 . j∈J In particular, for k ∈ J, c2k kϕk k4 ≤ Bkϕk k2 . Since there exist δ > 0 and j0 ∈ J such that kϕj k ≥ δ for all j ∈ J, j ≥ j0 , this implies SCALABLE FRAMES 7 ck ≤ B 1/2 δ −1 for all k ∈ J, k ≥ j0 . Thus Dc is bounded as kDc k = supj∈J ck .  2.2. General Equivalent Condition. We now state a seemingly obvious equivalent condition to scalability, which is however not straightforward to state and prove in the general setting of an arbitrary separable Hilbert space. Proposition 2.4. Let Φ = {ϕj }j∈J be a frame for H. Then the following conditions are equivalent. (i) Φ is (positively, strictly) scalable. (ii) There exists a non-negative (positive, strictly positive, respectively) diagonal operator D in ℓ2 (J) such that (2) TΦ∗ DDTΦ = IH . Proof. (i)⇒(ii). If Φ is scalable with a sequence of non-negative scalars (cj )j∈J , then Ψ := {cj ϕj }j∈J is a Parseval frame. In particular, Ψ is a frame, which, by Proposition 2.2, implies that ran TΦ ⊂ dom Dc and that SΨ = TΦ∗ Dc Dc TΦ is the frame operator of Ψ. Since the frame operator of a Parseval frame coincides with the identity operator, it follows that TΦ∗ Dc Dc TΦ = IH . (ii)⇒(i). Conversely, assume that there exists a non-negative diagonal operator D in ℓ2 (J) such that TΦ∗ DDTΦ = IH . Then DTΦ is everywhere defined. In particular, this implies that ran TΦ ⊂ dom D. Since TΦ is bounded and D is closed, the operator DTΦ is closed. Hence, by the closed graph theorem, DTΦ is a bounded operator from H into ℓ2 (J). In fact, (DTΦ )∗ (DTΦ ) = IH implies that DTΦ is even isometric. Thus, from the boundedness of TΦ we conclude that D| ran TΦ is ICR. Let c = (cj )j∈J be the sequence of non-negative scalars such that D = Dc . As a consequence of Proposition 2.2, Ψ := {cj ϕj }j∈J is a frame with frame operator SΨ = IH , which implies that Ψ is a Parseval frame. The proofs for positive and strict scalability of Φ follow analogous lines.  Under certain assumptions, the relation (2) can be simplified as stated in the following remark which directly follows from Corollary 2.3. Remark 2.5. If δ := lim inf j∈J kϕj k > 0, then a diagonal operator D as in Proposition 2.4 is necessarily bounded, and (2) reads TΦ∗ D 2 TΦ = IH . 8 G. KUTYNIOK, K. A. OKOUDJOU, F. PHILIPP, AND E. K. TULEY Before stating our main theorem in this section, we first provide a highly useful implication of Proposition 2.4, showing that scalability is invariant under unitary transformations. Corollary 2.6. Let U be a unitary operator in H. Then a frame Φ = {ϕj }j∈J for H is scalable if and only if the frame UΦ = {Uϕj }j∈J is scalable. Proof. Let Φ be a scalable frame for H with diagonal operator D. Since the analysis operator of UΦ is given by TU Φ = TΦ U ∗ , TU∗ Φ DDTU Φ = UTΦ∗ DDTΦ U ∗ = UTΦ∗ DDTΦ U ∗ = UU ∗ = IH , which implies scalability of UΦ. The converse direction can be proved similarly.  2.3. Main Result. To state the main result of this section, we require the notion of an orthogonal basis, which we recall for the convenience of the reader. A sequence {vk }k of non-zero vectors in a Hilbert space K is called an orthogonal basis of K, if inf k kvk k > 0 and (vk /kvk k)k is an orthonormal basis of K. The following result provides several equivalent conditions for a frame Φ to be strictly scalable. We are already familiar with condition (ii). Condition (iii) can be interpreted as a ‘diagonalization’ of the Grammian of Φ, and condition (iv) shows that Φ can be orthogonally expanded to an orthogonal basis. Theorem 2.7. Let Φ = {ϕj }j∈J be a frame for H such that lim inf j∈J kϕj k > 0, and let T = TΦ denote its analysis operator. Then the following statements are equivalent. (i) The frame Φ is strictly scalable. (ii) There exists a strictly positive bounded diagonal operator D in ℓ2 (J) such that DT is isometric (that is, T ∗ D 2 T = IH ). (iii) There exist a Hilbert space K and a bounded ICR operator L : K → ℓ2 (J) such that T T ∗ + LL∗ is a strictly positive bounded diagonal operator. (iv) There exist a Hilbert space K and a frame Ψ = {ψj }j∈J for K such that the vectors ϕj ⊕ ψj ∈ H ⊕ K, j ∈ J, form an orthogonal basis of H ⊕ K. If one of the above conditions holds, then the frame Ψ from (iv) is strictly scalable, its analysis operator is given by an operator L from (iii), and with a diagonal operator D from (ii) we have (3) L∗ D 2 L = IK , and L∗ D 2 T = 0. SCALABLE FRAMES 9 Proof. (i)⇔(ii). This equivalence follows from Proposition 2.4 (see also Remark 2.5). (ii)⇔(iii). For the proof of (ii)⇒(iii) let D be a strictly positive bounded diagonal operator in ℓ2 (J) such that T ∗ D 2 T = IH . For the Hilbert space K in (iii) we choose K := (ran DT )⊥ = ker T ∗ D ⊂ ℓ2 (J). On K we define the operator L : K → ℓ2 (J) by L := D −1 |K, which clearly is a bounded ICR operator. Then L∗ = PK D −1 , where PK denotes the orthogonal projection in ℓ2 (J) onto K. Let us show that DT T ∗D + PK coincides with the identity operator on ℓ2 (J). Then T T ∗ + LL∗ = D −1 DT T ∗ DD −1 + D −1 PK D −1
= D −1 DT T ∗ D + PK D −1 = D −2 , which is a strictly positive bounded diagonal operator in ℓ2 (J), and (iii) is proved. Since DT is isometric, we have
2 DT T ∗ D = DT (DT )∗(DT )T ∗D = DT T ∗ D, which shows that DT T ∗D is a projection. Moreover, DT T ∗D is selfadjoint and thus an orthogonal projection. Since its kernel coincides with ker T ∗ D = K, it is the orthogonal projection onto K⊥ . This shows that DT T ∗ D + PK = Iℓ2 (J) . To prove the converse implication, suppose that (iii) holds with a Hilbert space K and a bounded ICR operator L : K → ℓ(J), such that T T ∗ + LL∗ = D −2 with a strictly positive bounded diagonal operator D. Note that also D −1 is strictly positive and bounded. Define the operator     x x 2 (4) G : H ⊕ K → ℓ (J), G := T x + Ly, ∈ H ⊕ K. y y Then G∗ v = (T ∗ v, L∗ v)T , v ∈ ℓ2 (J), and hence GG∗ = T T ∗ + LL∗ = D −2 . In particular, G is an isomorphism between H ⊕ K and ℓ2 (J). Moreover, we have G∗ D 2 G = G∗ D 2 D −2 G−∗ = IH⊕K . This implies that  ∗ 2     ∗ T D T T ∗D2L IH 0 T 2 2 (D T, D L) = , = L∗ D 2 T L∗ D 2 L L∗ 0 IK or, equivalently, T ∗ D 2 T = IH , L∗ D 2 L = IK , which, in particular, yields (ii) (and (3)). and L∗ D 2 T = 0, 10 G. KUTYNIOK, K. A. OKOUDJOU, F. PHILIPP, AND E. K. TULEY (iii)⇔(iv). For the implication (iii)⇒(iv), let D, L and G be as above and define ψj := L∗ ej , j ∈ J, where ej denotes the j-th vector of the standard orthonormal basis {ej }j∈J of ℓ2 (J). As L is a bounded ICR operator and X X X |hx, ψj i|2 = |hx, L∗ ej i|2 = |hLx, ej i|2 = kLxk2 j∈J j∈J j∈J for all x ∈ K, it follows that Ψ = {ψj }j∈J is a frame. Note that T ∗ ej = ϕj , j ∈ J. Hence, ϕj ⊕ ψj = T ∗ ej ⊕ L∗ ej = G∗ ej , j ∈ J, and therefore hϕj ⊕ψj , ϕk ⊕ψk i = hG∗ ej , G∗ ek i = hGG∗ ej , ek i = hD −2ej , ek i = c−2 j δjk . As the cj ’s are bounded and G∗ is an isomorphism, this shows that the sequence {ϕj ⊕ ψj }j∈ J is an orthogonal basis of ℓ2 (J). Finally, to prove the converse implication, suppose that (iv) holds true and denote by L the analysis operator of the frame Ψ. Since {ϕj ⊕ ψj }j∈J is an orthogonal basis of H ⊕ K, for all j, k ∈ J we have hϕj , ϕk i + hψj , ψk i = dj δjk , where dj = kϕj k2 + kψj k2 , j ∈ J. Note that the sequence (dj )j∈J is bounded and bounded from below by a positive constant. Hence, for all j, k ∈ J, h(T T ∗ + LL∗ )ej , ek i = hT ∗ ej , T ∗ ek i + hL∗ ej , L∗ ek i = hϕj , ϕk i + hψj , ψk i = dj δjk = hdj ej , ek i. This implies T T ∗ + LL∗ = Dd , where d := (dj )j∈J . The operator Dd is a strictly positive bounded diagonal operator, which proves (iii).  The restriction of conditions (iii) and (iv) in Theorem 2.7 to the situation of finite frames is not immediate and requires some thought. This is the focus of the next result. N Corollary 2.8. Let Φ = {ϕj }M and let T = TΦ ∈ j=1 be a frame for K M ×N K denote the matrix representation of its analysis operator. Then the following statements are equivalent. (i) The frame Φ is strictly scalable. (ii) There exists a positive definite diagonal matrix D ∈ KM ×M such that DT is isometric. (iii) There exists L ∈ KM ×(M −N ) such that T T ∗ + LL∗ is a positive definite diagonal matrix. M −N (iv) There exists a frame Ψ = {ψj }M such that {ϕj ⊕ j=1 for K M M M ψj }j=1 ∈ K forms an orthogonal basis of K . Proof. We prove this result by using the equivalent conditions from Theorem 2.7. First of all, we observe that H = KN and ℓ2 (J) = KM . SCALABLE FRAMES 11 Moreover, condition (ii) obviously coincides with Theorem 2.7(ii), so that (i)⇔(ii) holds. The equivalence (iii)⇔(iv) can be shown in a similar way as the equivalence (iii)⇔(iv) in Theorem 2.7. Hence, it remains to show that (iii) and the condition (iii) in Theorem 2.7 are equivalent. For this, assume that (iii) holds, set K := KM −N and G := [T |L] ∈ KM ×M . Then, since GG∗ = T T ∗ + LL∗ is a positive definite diagonal matrix, it follows that G is non-singular and therefore ker L = {0}. Thus, L is ICR, and (iii) in Theorem 2.7 holds. For the converse, recall that the operator G : KN ⊕ K → KM in (4) was shown to be an isomorphism in the proof of Theorem 2.7. Hence, dim K = M −N. Thus, with some (bijective) isometry V : KM −N → K e := LV ∈ KM ×(M −N ) we have T T ∗ + L eL e∗ = T T ∗ + LL∗ . and L  Finally, we apply Theorem 2.7 to the special case of finite frames with N + 1 frame vectors in KN , which leads to a quite easily checkable condition for scalability. For this, we again require some prerequisites. N Letting Φ = {ϕi }M j=1 be a frame for the Hilbert space K , by OΦ we denote the set of indices k ∈ {1, . . . , M} for which hϕk , ϕj i = 0 holds for all j ∈ {1, . . . , M} \ {k}. Note that OΦ = {1, . . . , M} holds if and only if Φ is an orthogonal basis of KN . In particular, this implies M = N. +1 N Corollary 2.9. Let Φ = {ϕj }N such that ϕj 6= 0 j=1 be a frame for K for all j = 1, . . . , N + 1. Then OΦ 6= {1, . . . , N + 1}, and the following statements are equivalent. (i) Φ is strictly scalable. (ii) There exist k ∈ {1, . . . , N + 1} \ OΦ and c > 0 such that hϕi , ϕk ihϕk , ϕj i = −chϕi , ϕj i holds for all i, j ∈ {1, . . . , N + 1} \ {k}, i 6= j. (iii) For all k ∈ {1, . . . , N + 1} \ OΦ there exists ck > 0 such that hϕi , ϕk ihϕk , ϕj i = −ck hϕi , ϕj i holds for all i, j ∈ {1, . . . , N + 1} \ {k}, i 6= j. Proof. As remarked before, OΦ = {1, . . . , N + 1} implies that Φ is an orthogonal basis of KN , which is impossible. (i)⇒(iii). For this, let k ∈ {1, . . . , N + 1} \ OΦ be arbitrary. By Theorem 2.7 (see also Corollary 2.8) there exists v = (v1 , . . . , vN +1 )T ∈ KN +1 such that TΦ TΦ∗ + vv ∗ is a diagonal matrix. Hence, hϕi , ϕj i + vi vj = 0 holds for all i, j ∈ {1, . . . , N + 1}, i 6= j. Therefore, for i, j ∈ {1, . . . , N + 1} \ {k}, i 6= j, we have hϕi , ϕk ihϕk , ϕj i = vi vj |vk |2 = −|vk |2 hϕi , ϕj i. 12 G. KUTYNIOK, K. A. OKOUDJOU, F. PHILIPP, AND E. K. TULEY If vk = 0, then hϕk , ϕj i = 0 for all j ∈ {1, . . . , N + 1} \ {k}. But since k∈ / OΦ was assumed, it follows that |vk |2 > 0, and (iii) holds. (iii)⇒(ii). This is obvious. (ii)⇒(i). Assume now that (ii) is satisfied, and set √ vk := c and vj := −vk−1 hϕj , ϕk i (j ∈ {1, . . . , N + 1} \ {k}). Then vi vk = −hϕi , ϕk i for i ∈ {1, . . . , N + 1} \ {k} and vi vj = |vk |−2 hϕi , ϕk ihϕk , ϕj i = −hϕi , ϕj i for i, j ∈ {1, . . . , N + 1} \ {k}, i 6= j. This implies that TΦ TΦ∗ + vv ∗ is a diagonal matrix whose diagonal entries are positive (since otherwise 0 = kϕj k2 + |vj |2 and thus ϕj = 0 for some j ∈ {1, . . . , N + 1}). Now, (i) follows from Theorem 2.7.  As mentioned above, Corollary 2.9 might be utilized to test whether a frame for KN with N + 1 frame vectors is strictly scalable or not. Such a test would consist of finding an index k ∈ / OΦ and checking whether there exists a c > 0 such that hϕi , ϕk ihϕk , ϕj i = −chϕi , ϕj i holds for all i, j ∈ {1, . . . , N + 1} \ {k}, i 6= j. 3. Scalability of Real Finite Frames We next aim for a more geometric characterization of scalability. For this, we now focus on frames for RN . The reason why we restrict ourselves to real frames is that in the proof of the main theorem in this section we make use of the following variant of Farkas’ Lemma which only exists for real vector spaces. Lemma 3.1. Let A : V → W be a linear mapping between finitedimensional real Hilbert spaces (V, h· , ·iV ) and (W, h· , ·iW ), let {ei }N i=1 be an orthonormal basis of V and let b ∈ W . Then exactly one of the following statements holds: (i) There exists x ∈ V such that Ax = b and hx, ei iV ≥ 0 for all i = 1, . . . , N. (ii) There exists y ∈ W such that hb, yiW < 0 and hAei , yiW ≥ 0 for all i = 1, . . . , N. Lemma 3.1 can be proved in complete analogy to the classical Farkas’ Lemma, where V = Rn and W = Rm , n, m ∈ N. A proof of this statement can, for instance, be found in [3, Thm 5.1]. 3.1. Characterization Result. The following theorem provides a characterization of non-scalability of a finite frame specifically tailored to the finite-dimensional case. In Subsection 3.2, condition (iii) will then be utilized to derive an illuminating geometric interpretation. SCALABLE FRAMES 13 N N Theorem 3.2. Let Φ = {ϕj }M j=1 ⊂ R \ {0} be a frame for R . Then the following statements are equivalent. (i) Φ is not scalable. (ii) There exists a symmetric matrix Y ∈ RN ×N with tr(Y ) < 0 such that ϕTj Y ϕj ≥ 0 for all j = 1, . . . , M. (iii) There exists a symmetric matrix Y ∈ RN ×N with tr(Y ) = 0 such that ϕTj Y ϕj > 0 for all j = 1, . . . , M. Proof. (i)⇔(ii). Let W denote the vector space of all symmetric matrices X ∈ RN ×N , and let h· , ·iW denote the scalar product on W defined by hX, Y iW := tr(XY ), X, Y ∈ W . Furthermore, define the linear mapping A : RM → W by Ax := TΦT diag(x)TΦ , x ∈ RM . By Proposition 2.4 the frame Φ is not scalable if and only if there exists no x ∈ RM , x ≥ 0, with Ax = IN . Hence, due to Lemma 3.1, Φ is not scalable if and only if there exists Y ∈ W with tr(Y ) = hIN , Y iW < 0 such that 0 ≤ hAej , Y iW = tr((Aej )Y ) = tr(ϕj ϕTj Y ) = ϕTj Y ϕj holds for all j = 1, . . . , M, where {ej }M j=1 denotes the standard basis of M R . This proves the equivalence of (i) and (ii). (ii)⇒(iii). For this, let Y1 ∈ W with α := − tr(Y1 ) > 0 such that T ϕj Y1 ϕj ≥ 0 for all j = 1, . . . , M, and set Y := Y1 + Nα IN . Then tr(Y ) = 0 and ϕTj Y ϕj > 0 for all j = 1, . . . , M, as desired. (iii)⇒(i). Assume now, that there exists Y ∈ W as in (iii), that is, hIN , Y iW = 0 and hAej , Y iW > 0 for all j. Suppose that Φ is scalable. Then there exists x ∈ RM , x ≥ 0, such that Ax = IN . This implies 0 = hIN , Y iW = hAx, Y iW = M X j=1 xj hAej , Y iW , which yields x = 0, contrary to the assumption Ax = IN . The theorem is proved.  This theorem can be used to derive a result on the topological structure of the set of non-scalable frames for RN . In fact, the corollary we will draw shows that this set is open in the following sense. N N Corollary 3.3. Let Φ = {ϕj }M which j=1 ⊂ R \ {0} be a frame for R is not scalable. Then there exists ε > 0 such that each set of vectors N {ψj }M with j=1 ⊂ R (5) kϕj − ψj k < ε for all j = 1, . . . , M 14 G. KUTYNIOK, K. A. OKOUDJOU, F. PHILIPP, AND E. K. TULEY is a frame for RN which is not scalable. Proof. Choosing a subset J of {1, . . . , M} such that {ϕj }j∈J is a basis of RN , it follows from the continuity of the determinant that there N exists ε1 > 0 such that all sets of vectors {ψj }M with (5) (ε j=1 ⊂ R replaced by ε1 ) are frames. By Theorem 3.2, there exists a symmetric matrix Y ∈ RN ×N with tr(Y ) < 0 such that ϕTi Y ϕi ≥ 0 for all i. By adding δIN to Y with some δ > 0 we may assume without loss of generality that tr(Y ) < 0 and ϕTj Y ϕj > 0 for all j (note that the frame vectors of Φ are assumed to be non-zero). Since the function x 7→ xT Y x is continuous, it follows that there exists ε ∈ (0, ε1) such N T that for each frame {ψj }M j=1 ⊂ R with (5) we have ψj Y ψj > 0 for all j. By Theorem 3.2, the frame {ψj }M j=1 is not scalable, which finishes the proof.  3.2. Geometric Interpretation. We now aim to analyze the geometry of the vectors of a non-scalable frame. To derive a precise geometric characterization of non-scalability, we will in particular exploit Theorem 3.2. As a first step, notice that each of the sets C± (Y ) := {x ∈ RN : ±xT Y x > 0}, Y ∈ RN ×N symmetric, considered in Theorem 3.2 (iii) is in fact an open cone with the additional property that x ∈ C± (Y ) implies −x ∈ C± (Y ). Thus, in the sequel we need to focus our attention on the impact of the condition tr(Y ) = 0 on the shape of these cones. We start by introducing a particular class of conical surfaces, which due to their relation to quadrics – the exact relation being revealed below – are coined ‘conical zero-trace quadrics’. Definition 3.4. Let the class of conical zero-trace quadrics CN be defined as the family of sets ) ( N −1 X ak hx, ek i2 = hx, eN i2 , (6) x ∈ RN : k=1 N −1 where runs through all orthonormal bases of RN and (ak )k=1 PN −1 runs through all tuples of elements in R \ {0} with k=1 ak = 1. {ek }N k=1 The next example provides some intuition on the geometry of the elements in this class in dimension N = 2, 3. SCALABLE FRAMES 15 Example 3.5. √ • N = 2. In this case, by setting e± := (1/ 2)(e1 ±e2 ), a straightforward computation shows that C2 is the family of sets {x ∈ R2 : hx, e− ihx, e+ i = 0}, where {e− , e+ } runs through all orthonormal bases of R2 . Thus, each set in C2 is the boundary surface of a quadrant cone in R2 , i.e., the union of two orthogonal one-dimensional subspaces in R2 . • N = 3. In this case, it is not difficult to prove that C2 is the family of sets  x ∈ R3 : ahx, e1 i2 + (1 − a)hx, e2 i2 = hx, e3 i2 , where {ei }3i=1 runs through all orthonormal bases of R3 and a runs through all elements in (0, 1). The sets in C3 are the boundary surfaces of a particular class of elliptical cones in R3 . To analyze the structure of these conical surfaces we let {e1 , e2 , e3 } be the standard unit basis and a ∈ (0, 1). Then the quadric  x ∈ R3 : ahx, e1 i2 + (1 − a)hx, e2 i2 = hx, e3 i2 intersects the planes {x3 = ±1} in  (x1 , x2 , ±1) : ax21 + (1 − a)x22 = 1 . These two sets are ellipses whose union contains the corner points (±1, ±1, ±1) of the unit cube. Thus, the considered quadrics are elliptical conical surfaces with their vertex in the origin, characterized by the fact that they intersect the corners of a rotated unit cube in R3 , see also Figure 3.2(b) and (c). Note that (6) is by rotation unitarily equivalent to the set ( ) N −1 X (7) x ∈ RN : x2N − ak x2k = 0 . k=1 Such surfaces uniquely determine cones by considering their interior or exterior. Similarly, we call the sets ) ( N −1 X ak hx, ek i2 < hx, eN i2 x ∈ RN : and ( k=1 x ∈ RN : N −1 X k=1 ak hx, ek i2 > hx, eN i2 ) the interior and the exterior of the conical zero-trace quadric in (6), respectively. 16 G. KUTYNIOK, K. A. OKOUDJOU, F. PHILIPP, AND E. K. TULEY Armed with this notion, we can now state the result on the geometric characterization of non-scalability. Theorem 3.6. Let Φ ⊂ RN \{0} be a frame for RN . Then the following conditions are equivalent. (i) Φ is not scalable. (ii) All frame vectors of Φ are contained in the interior of a conical zero-trace quadric of CN . (iii) All frame vectors of Φ are contained in the exterior of a conical zero-trace quadric of CN . Proof. We only prove (i)⇔(ii). The equivalence (i)⇔(iii) can be proved N N similarly. By Theorem 3.2, a frame Φ = {ϕj }M j=1 ⊂ R \ {0} for R is not scalable if and only if there exists a real symmetric N ×N-matrix Y with tr(Y ) = 0 such that ϕTj Y ϕj > 0 for all j = 1, . . . , M. Equivalently, there exist an orthogonal matrix U ∈ RN ×N and a diagonal matrix D ∈ RN ×N with tr(D) = 0 such that (Uϕj )T D(Uϕj ) > 0 for all j = 1, . . . , M. Note that, due to continuity reasons, the matrix D can be chosen non-singular, i.e., without zero-entries on the diagonal. Hence, N (i) is equivalent to the existence of an orthonormal basis {ek }N k=1 of R PN and values d1 , . . . , dN ∈ R \ {0} satisfying k=1 dk = 0 and N X k=1 dk hϕj , ek i2 > 0 for all j = 1, . . . , M. By a permutation of {1, . . . , N} we can achieve that dN > 0. Hence, by setting ak := −dk /dN for k = 1, . . . , N − 1, we see that (i) holds N if and only if there exist an orthonormal basis {ek }N and k=1 of R PN −1 a1 , . . . , aN −1 ∈ R \ {0} such that k=1 ak = 1 and N −1 X k=1 ak hϕj , ek i2 < hϕj , eN i2 But this is equivalent to (ii). for all j = 1, . . . , M.  ∗ By CN we denote the subclass of CN consisting of all zero-trace conical quadrics in which the orthonormal basis is the standard basis of RN . ∗ That is, the elements of CN are quadrics of the form (7) with nonPN −1 zero ak ’s satisfying k=1 ak = 1. The next corollary is an immediate consequence of Theorem 3.6 and Corollary 2.6. Corollary 3.7. Let Φ ⊂ RN \ {0} be a frame for RN . Then the following conditions are equivalent. (i) Φ is not scalable. SCALABLE FRAMES (ii) There exists an orthogonal matrix U ∈ RN ×N tors of UΦ are contained in the interior of a ∗ quadric of CN . (iii) There exists an orthogonal matrix U ∈ RN ×N tors of UΦ are contained in the exterior of a ∗ quadric of CN . 17 such that all vecconical zero-trace such that all vecconical zero-trace Utilizing Example 3.5, we can draw the following conclusion from Theorem 3.6 for the cases N = 2, 3. Corollary 3.8. (i) A frame Φ ⊂ R2 \ {0} for R2 is not scalable if and only if there exists an open quadrant cone which contains all frame vectors of Φ. (ii) A frame Φ ⊂ R3 \ {0} for R3 is not scalable if and only if all frame vectors of Φ are contained in the interior of an elliptical conical surface with vertex 0 and intersecting the corners of a rotated unit cube. To illustrate the geometric characterization, Figure 3.2 shows sample regions of vectors of a non-scalable frame in R2 and R3 . (a) (b) (c) Figure 1. (a) shows a sample region of vectors of a nonscalable frame in R2 . (b) and (c) show examples of C3− and C3+ which determine sample regions in R3 . 4. Acknowledgements G. Kutyniok acknowledges support by the Einstein Foundation Berlin, by Deutsche Forschungsgemeinschaft (DFG) Grant SPP-1324 KU 1446/13 and DFG Grant KU 1446/14, and by the DFG Research Center Matheon “Mathematics for key technologies” in Berlin. F. Philipp is supported by the DFG Research Center Matheon. K. A. Okoudjou was supported by ONR grants N000140910324 and N000140910144, by a RASA from the Graduate School of UMCP and by the Alexander von Humboldt foundation. He would also like to express his gratitude to 18 G. KUTYNIOK, K. A. OKOUDJOU, F. PHILIPP, AND E. K. TULEY the Institute for Mathematics at the University of Osnabrück for its hospitality while part of this work was completed. References [1] P. Balazs, J.-P. Antoine, and A. Grybos, Weighted and controlled frames: Mutual relationship and first numerical properties, Int. J. Wavelets Multiresolut. Inf. Process. 8 (1) (2010), 109–132. [2] V. Balakrishnan, and S. Boyd, Existence and uniqueness of optimal matrix scalings, SIAM J. 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[16] A. Rahimi and P. Balazs, Multipliers for p-Bessel sequences in Banach spaces, Integral Equations Oper. Theory 68 (2) (2010), 193–205. [17] D. Ruiz, A scaling algorithm to equilibrate both rows and columns norms in matrices, Technical Report RT/APO/01/4, ENSEEIHT-IRIT, 2001. [18] A. Shapiro, Upper bounds for nearly optimal diagonal scaling of matrices, Linear Multilinear Algebra 29 (2) (1991), 145–147. [19] A. Shapiro, Optimal block diagonal l2 -scaling of matrices, SIAM J. Numer. Anal. 22 (1985), 81–94. SCALABLE FRAMES 19 [20] A. van der Sluis, Condition numbers and equilibration of matrices, Numer. Math. 14 (1969), 14-23. Gitta Kutyniok, Institut für Mathematik, Technische Universität Berlin, Strasse des 17. Juni 136, 10623 Berlin, Germany E-mail address: kutyniok@math.tu-berlin.de Kasso A. Okoudjou, Department of Mathematics, University of Maryland, College Park, MD 20742 USA E-mail address: kasso@math.umd.edu Technische Universität Berlin, Institut für Mathematik, 10623 Berlin, Germany E-mail address: philipp@math.tu-berlin.de Elizabeth K. Tuley, Department of Mathematics, University of California at Los Angeles, Los Angeles, CA 90095 USA E-mail address: ektuley@umd.edu